Abstract
We consider the capacity of entanglement in models related with the gravitational phase transitions. The capacity is labeled by the replica parameter which plays a similar role to the inverse temperature in thermodynamics. In the end of the world brane model of a radiating black hole the capacity has a peak around the Page time indicating the phase transition between replica wormhole geometries of different types of topology. Similarly, in a moving mirror model describing Hawking radiation the capacity typically shows a discontinuity when the dominant saddle switches between two phases, which can be seen as a formation of island regions. In either case we find the capacity can be an invaluable diagnostic for a black hole evaporation process.
Highlights
JHEP05(2021)062 manifestation of the generalized gravitational entropy [9] which derives from a gravitational path integral using the replica method for entanglement entropy: SA
The most general formulation [14] expands on the notion of the generalized second law [15] and states that the entropy of the boundary region A is given by the generalized entropy consisting of the area of a minimal quantum extremal surface (QES) and the entanglement entropy of a matter across the QES
The Ryu-Takanayagi formula picks up only the replica geometry with disconnected replicas while the island formula has a substantial contribution from the replica wormhole with all replicas connected in gravitational regions, which instructs us to include a region, named an island, behind a black hole horizon as a part of the entangling region in calculating the entanglement entropy of the Hawking radiation
Summary
We will consider a toy 2d gravity model of an evaporating black hole in the Jackiw-Teitelboim (JT) gravity [83, 84] with an end of the world (EOW) brane entangled with an auxiliary system [18] In this simple model, island contributions to entanglement entropy in the auxiliary system are obtained from replica wormhole contributions to the. For large dimension of the systems, the replica wormholes contributions can be calculated by summing up all planar topologies in the full gravitational path-integral as performed in [18] It can be done analytically in the microcanonical ensemble and numerically in the canonical ensemble. We will find smooth curves of the capacity
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